## Monday, 21 December 2020

### Maxwell's equations have something missing

 James C. Maxwell is peeved
I return to chapter one and Maxwell's equations.  Carroll's version of Maxwell's equations "in 19th century notation" at his equations 1.92 are$$\nabla\times\mathbf{B}-\partial_t\mathbf{E}=\mathbf{J}$$
$$\nabla\bullet\mathbf{E}=\rho$$
$$\nabla\times\mathbf{E}+\partial_t\mathbf{B}=0$$
$$\nabla\bullet\mathbf{B}=0$$
But that's not Maxwell's equations! In SI units they are

$$\nabla\times\mathbf{B}-\frac{1}{c^2}\partial_t\mathbf{E}=\mu_0\mathbf{J}$$
$$\nabla\bullet\mathbf{E}=\frac{\rho}{\epsilon_0}$$
$$\nabla\times\mathbf{E}+\partial_t\mathbf{B}=0$$
$$\nabla\bullet\mathbf{B}=0$$
Where $\epsilon_0,\mu_0$ are the electric and magnetic constants. Carroll has already announced that we were setting the speed of light $c=1$ but he does not say anything about $\epsilon_0,\mu_0$ here or later. They are just left out. I first noticed this when calculating the Energy-Momentum tensor for electro-magnetic radiation. I think that in Carrol's "natural units" (which might be a bit unorthodox) we also have $\epsilon_0=1$ and that means that, as $c=1$, $\mu_0=1$ too.

See my reasoning in Commentary 1.8 Maxwells equations and units.pdf (two pages).